Fuzzy Partition and Correlation for Image Segmentation ... · Fuzzy Partition and Correlation for Image Segmentation with Differential Evolution Fangyan Nie, Member, IAENG, and Pingfeng

Fuzzy Partition and Correlation for ImageSegmentation with Differential Evolution

Fangyan Nie, Member, IAENG, and Pingfeng Zhang

Abstract—Thresholding-based techniques have been widelyused in image segmentation. The selection of appropriatethreshold is a very significant issue for image thresholding.In this paper, a new image histogram thresholding methodbased on fuzzy partition and maximum correlation criterion ispresented. In the proposed approach, the regions, i.e. object andbackground, are considered ambiguous in nature, and hence theregions are transformed into fuzzy domain with membershipfunctions. Then, the fuzzy correlations about regions are con-structed and the optimal threshold is determined by searchingan optimal parameter combination of the membership functionssuch that the correlation of the fuzzy partitions is maximized. S-ince the exhaustive search for all fuzzy parameter combinationsis too costly, the differential evolution algorithm is introducedinto fuzzy correlation image segmentation to solve this optimalproblem adaptively. Experimental results on general images andinfrared images demonstrate the effectiveness of the proposedmethod.

Index Terms—image thresholding, fuzzy partition, maximumcorrelation criterion, differential evolution algorithm

I. INTRODUCTION

IMAGE segmentation is an important low-level prepro-cessing step for many computer vision problems. The pur-

pose of this step is that objects and background are separatedinto nonoverlapping sets [1], [2]. Usually, this segmentationprocess is based on the image gray-level histogram, namelyimage histogram thresholding [3]. In that case, the aim isto find a critical value or threshold. Through this threshold,applied to the whole image, pixels whose gray levels exceedthis critical value are assigned to one set and the rest to theother. In recent decades, many thresholding methods havebeen proposed [4]–[23]. In 1979, Otsu proposed a methodthat maximizes the separability of the resultant classes ingray levels utilizing a between-class variance function [4].The Otsu method is one of the most famous thresholdingapproaches for image segmentation. Entropy model is alsoone of the most popular techniques in thresholding [5]–[10].The Kapur method is developed based on the maximizationof the class entropies [6]. Inspired by the idea of chaos andfractal theory, Yen et al. [12] presented a novel criterionfor image thresholding, i.e. maximum correlation criterion.Computational analyses and simulation results indicate thehigh effectiveness of the maximum correlation criterion forimage thresholding [12].

Manuscript received March 27, 2013; revised July 10, 2013. This workwas supported in part by the Young Core Instructor Foundation of HunanProvincial Institutions of Higher Education of China and the Doctor Scien-tific Research Startup Project Foundation of Hunan University of Arts andScience.

Fangyan Nie is with the Institute of Graphics and Image ProcessingTechnology, Hunan University of Arts and Science, Changde 415000, China.e-mail: [email protected].

Pingfeng Zhang is with the College of Computer Science and Technology,Hunan University of Arts and Science, Changde 415000, China.

There are some fuzziness factors in nature on imageprocessing such as information loss while mapping 3-Dobjects into 2-D images, ambiguity and vagueness in somedefinitions (such as edges, boundaries, regions, and textures),ambiguity and vagueness in interpreting low-level imageprocessing results [13]–[15]. Fuzzy sets theory is findingextensive applications to describe the vague concepts in themodern mathematical framework. For example, in patternclassification problem, instead of the conventional determin-istic assignment of a sample to a class, fuzzy partitioningstrategies provide soft description of the classes, where eachof the sample points is assigned a membership in eachof the classes. Image thresholding can be regarded as aclassic pixels classification problem. To tackle the fuzzycharacteristics exists in images, several fuzzy thresholdingmethods have been proposed for image segmentation, suchas fuzzy partition entropy method [14]–[18].

In this paper, we present a new thresholding methodutilizing the maximum correlation criterion based on theprobability partition, fuzzy partition and differential evolu-tion (DE) [24] algorithm. The image is divided into twoparts: object and background in the proposed method, andthe Z-function and S-function are adopted as the membershipfunctions of background and object. Since the exhaustivesearch for all fuzzy parameter combinations of memberfunctions is too costly, we introduced DE algorithm intofuzzy image segmentation to solve this optimal problemadaptively.

The paper is organized as follows. In Section 2, themaximum correlation criterion is reviewed and the fuzzy cor-relation criterion is proposed based on probability analysis,fuzzy partition and correlation theory. In Section 3, the basicprinciple of DE is described and the specific method of DEapplied in fuzzy correlation image segmentation is proposed.More experiments are discussed in Section 4 to demonstratethe effectiveness and usefulness of the proposed approach.Finally, the conclusions are drawn in Section 5.

II. FUZZY PARTITION AND MAXIMUM CORRELATION FORIMAGE SEGMENTATION

In this section, we give brief descriptions about the corre-lation and fuzzy partition approach for image segmentation.

A. Maximum correlation criterionLet I = [f(x, y)]M×N be an L-level digital image,

f(x, y) ∈ G, G = {0, 1, 2, · · · , L − 1} is the set ofgray levels. Let hi be the frequency of gray level i andpi = hi/(M ×N) be the probability of occurrence of graylevel i. Hence, a distribution {pi|i ∈ G} can be obtained.Suppose t is an assumed threshold, t partitions the image intotwo regions, namely, the object O and the background B. Weassume gray values in [t+1, · · · , L−1] constitute the object

IAENG International Journal of Computer Science, 40:3, IJCS_40_3_03

(Advance online publication: 19 August 2013)

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region, while those in [0, · · · , t] constitute the background. If∑ti=0 pi is larger than zero and smaller than one, then two

distributions can be derived from distribution {pi|i ∈ G}after normalization.

B =

{p0P (t)

,p1P (t)

, · · · , ptP (t)

}O =

{pt+1

1− P (t),

pt+2

1− P (t), · · · , pL−1

1− P (t)

} (1)

Where P (t) =∑t

i=0 pi is the total probability up to thetth gray level. In the maximum correlation criterion, thebasic idea is to choose the threshold such that the totalamount of correlation provided by the object and backgroundis maximized. The total amount of correlation provided bydistributions B and O is [12]

TC(t) = CB(t) + CO(t)

= − ln

t∑i=0

(pi

P (t)

)2

− ln

L−1∑i=t+1

(pi

1− P (t)

)2 (2)

In order to obtain the maximal correlation contributed bythe object and background in the image I , TC(t) must bemaximized. The maximum correlation criterion [12] it todetermine the threshold t∗ such that

TC(t∗) = argmaxt∈G

[TC(t)] (3)

B. Fuzzy partition

The concept of fuzzy partitioning can be extended fordigital image thresholding by visualizing the object andbackground regions as fuzzy sets, O and B, with each ofthe pixels showing a partial membership in each of theregions according to its gray value, i.e. µO(g) ∈ [0, 1],µB(g) ∈ [0, 1]. With this sort of a partition, regions are nomore guaranteed to be mutually exclusive; in other words,there may not exist a crisp boundary between regions.

A fuzzy thresholded description of an image can becharacterized by two membership functions µO and µB

in such a way that they reflect the nature of object andbackground gray distribution event after thresholding. Thedescription reflects the compatibility measure of each of thepixel/gray value in object and background regions.

C. Fuzzy correlation for image thresholding

In the proposed fuzzy maximum correlation approach, thethresholding is used to classify pixels to object group andbackground group. With this aim, two fuzzy sets, objectO and background B be considered, whose membershipfunctions are defined in this paper as below

µB(g) =

1, g ≤ a

1− (g − a)2

(c− a)(b− a), a < g ≤ b

(g − c)2

(c− a)(c− b), b < g ≤ c

0, g > c

(4)

µO(g) =

0, g ≤ a

(g − a)2

(c− a)(b− a), a < g ≤ b

1− (g − c)2

(c− a)(c− b), b < g ≤ c

1, g > c

(5)

Where 0 ≤ a < b < c ≤ L − 1, g ∈ G is the independentvariable, a, b, and c are parameters determining the shape ofthe above two membership functions as shown in Figure 1.Obviously, µB(g) = 1−µO(g). Hence, µB(g) and µO(g) isfuzzy 2-partition of the image. The probabilities of the twofuzzy events object O and background B are defined as

PB =

L−1∑g=0

µB(g)pg (6)

PO =

L−1∑g=0

µO(g)pg (7)

According to maximum correlation criterion [12] for im-age thresholding, in the proposed approach we define theamount of fuzzy correlation provided by fuzzy events objectand background as follows

FCB = − ln

L−1∑g=0

(µB(g)pg

PB

)(8)

FCO = − ln

L−1∑g=0

(µO(g)pg

PO

)(9)

The total amount of fuzzy correlation is defined as

FC(a, b, c) = FCB + FCO (10)

Since different combination of (a, b, c) corresponds todifferent fuzzy 2-partition, so, the fuzzy correlation variesalong with three parameters a, b, and c. We can find an op-timal combination of (a, b, c) such that the fuzzy correlationFC(a, b, c) has maximum value, i.e.

(a∗, b∗, c∗) = argmax [FC(a, b, c)] (11)

Then, the optimal threshold t∗ for image segmentation canbe computed as

µB(t) = µO(t) (12)

As shown in Figure 1, threshold t∗ is the point of inter-section of µB and µO. According to Eqs. (4) and (5), thesolution can be computed as

t∗ =

a∗ +

√(c∗ − a∗)(b∗ − a∗)

2, (a∗ + c∗)/2 ≤ b∗ ≤ c∗

c∗ −√

(c∗ − a∗)(c∗ − b∗)

2, a∗ ≤ b∗ ≤ (a∗ + c∗)/2

(13)



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Fig. 1: The plot of membership function with (a, b, c) = (15, 120, 245) and the optimal threshold t∗

III. FUZZY PARTITION AND MAXIMUM CORRELATIONFOR IMAGE SEGMENTATION

From above description, we can see that it is not an easywork for finding the optimal fuzzy membership functionparameters combination of (a, b, c). Since a, b, and c maytake values from {0, 1, · · · , L− 1}, the search space of pa-rameters combination (a, b, c) is close to L3. Such procedureis characterized by high computation cost. In this section, wepropose a DE algorithm to find the optimal combination of(a, b, c).

A. Description of differential evolutionDifferential evolution is an evolutionary computation al-

gorithm developed by Storn and Price which solves realvalued problems based on the principles of natural evolution[24]. It is a heuristic optimization method which can beused to optimize nonlinear and non-dfferentiable continuousspace functions. It has been extended to handle mixed integerdiscrete continuous optimization problem also [23]–[26]. DEuses a population P of size n that evolves over t generationsto reach the optimal solution. Each individual Xi is a vectorthat contains as many parameters as the problem decisionvariables D.

P t =[Xt

1, Xt2, · · · , Xt

n

](14)

Xti =

[Xt

i,1, Xti,2, · · · , Xt

i,D

], i = 1, 2, · · · , n (15)

The population size n is an algorithm control parameterselected by the user which remains constant throughoutthe optimization process. The optimization process in DEis carried out using the three basic operations: Mutation,Crossover and Selection. The algorithm starts by creating aninitial population of n vectors. Random values are assignedto each decision parameter in every vector as follows.

X0i,j = XL

j + r ·(XU

j −XLj

)(16)

Where i = 1, 2, · · · , n and j = 1, 2, · · · , D; XLj and XU

j arethe lower and upper bounds of the jth decision parameter;and r is a uniformly distributed random number within [0, 1]generated for each value of j. X0

i,j is the jth parameter ofthe ith individual of the initial population. The main stepsof the DE algorithm are shown in Figure 2.

In DE algorithm, the mutation operator creates mutantvectors Di by perturbing a randomly selected vector Xa with

Initialization EvaluationRepeat

MutationCrossoverEvaluationSelection

Until (termination criteria are met)

Fig. 2: Illustration of the main steps of the DE algorithm

the difference of two other randomly selected vectors Xb andXc.

Di = Xa + F · (Xa −Xc), i = 1, 2, · · · , n (17)

Where Xa, Xb, and Xc are randomly chosen vectors amongthe n population, and a = b = c = i. The scaling constantF is an algorithm control parameter used to adjust theperturbation size in the mutation operator and to improvealgorithm convergence.

The crossover operation generates trail vectors Ti by mix-ing the parameters of the mutant vectors Di with the targetvector Xi according to a selected probability distribution.

Ti,j =

{Di,j , if((rn ≤ CR)or(j = d))

Xi,j , otherwise(18)

Where i = 1, 2, · · · , n and j = 1, 2, · · · , D; d is a randomlychosen index from {1, 2, · · · , D} that guarantees that thetrail vector gets at least one parameter from the mutantvector; rn is a uniformly distributed random number within[0, 1] generated for each value of j. The crossover constantCR is an algorithm parameter that controls the diversity ofthe population and aids the algorithm to escape from localminima. Xi,j , Di,j and Ti,j are the jth parameter of theith target vector, mutant vector and trail vector at currentgeneration, respectively.

The selection operation forms the population by choos-ing between the trail vectors and their predecessors (targetvectors) those individuals that present a better fitness or are



______________________________________________________________________________________

more optimal according to Eq. (19).

Xt+1i =

{Ti, if(f(Ti) > f(Xt

i ))

Xti , otherwise

(19)

This process is repeated for several generations allowingindividuals to improve their fitness as they explore thesolution space in search of optimal values.

DE has three essential control parameters: the scalingfactor F , the crossover constant CR and the populationsize n. The scaling factor is a value in the range [0, 2] thatcontrols the amount of perturbation in the mutation process.The crossover constant is a value in the range [0, 1] thatcontrols the diversity of the population. The population sizedetermines the number of individuals in the population andprovides the algorithm enough diversity to search the solutionspace.

B. DE implementation for image segmentationWhile applying DE to solve the parameters optimization

problem of fuzzy membership function, each vector in theDE population represents a candidate solution of the givenproblem. Considering the optimal problem mentioned above,the individuals in the population are constructed and thepopulation search strategy is proposed. We choose 3 asthe dimension of the search space, i.e. D = 3 and eachindividual in the population is three-dimensional vector ofreal numbers. The fuzzy correlation function (10) is the givenfitness function. The procedure is described as follows.

Step 1. Initialize n individuals in population according Eq.(16), where XL

j and XUj are the minimum and maximum

gray level of the given image, respectively.Step 2. Evaluate the fitness of each individual in the

population using the fuzzy correlation function (10).Step 3. Carry out the mutation, crossover, evaluation and

selection operators for each individual in the population. Inorder to obtain better convergence, the scaling factor F andthe crossover constant CR are set as follows

F = 0.5× (1 + rand(0, 1)) (20)

CR = CRmin+(CRmax−CRmin) ·(Itmax−It)/It (21)

Where rand(0,1) denotes a uniform random number withrange [0, 1]; CRmax and CRmin denote the maximum andminimum of cross probability CR, they can be assignedvalue in the step of initialization; Itmax and It denote thebiggest iteration number that be allowed and the currentiteration number of the DE algorithm, respectively.

Step 4. Repeat Step 3 until termination criteria are met.

IV. EXPERIMENTAL RESULTS AND ANALYSISFor evaluation the performance of the proposed approach,

we have applied the proposed method to a variety of imagesand compared the performance with that of some existingmethods. The thresholding methods proposed in references[16], [17] and [18] are based on fuzzy set which claimed thesatisfactory performance in image segmentation. Therefore,we compared the performance of the proposed algorithmwith that of the method in references [16], [17] and [18]. Forthe sake of convenience, we call these methods in references[16], [17] and [18] Benabdelkader method, Tao method andTang method, respectively. In addition, we compared the

performance of the proposed method with that of someclassical image thresholding methods, i.e. Otsu method [4],Pun entropy method [5], Kapur entropy method [6], and Yencorrelation method [12].

(a) (b)

(c) (d)

Fig. 3: The test images. Original image of (a) tire, (b) plane,(c) milkdrop and (d) shot1

(a) (b)

(c) (d)

Fig. 4: The histograms of test images. (a) tire, (b) plane, (c)milkdrop and (d) shot1

The all algorithms were coded in Matlab version 7 andrun on an Intel 2.66 GHz Pentium4 CPU, 2.00 GB RAMpersonal computer, under Microsoft Windows XP pro Oper-ating System. In our experiments the population size of theproposed DE algorithm is set to 30, the number of maximumiteration Itmax, the maximum cross probability CRmax andthe minimum cross probability CRmin are set to 100, 0.9and 0.1, respectively.

Firstly, four real-world images with different shapes of



______________________________________________________________________________________

gray levels histogram distribution are applied to test thesegmentation performance of different thresholding methods.The four images are ‘tire’ image, ‘plane’ image, ‘milkdrop’image and ‘shot1’ image and their sizes are 232 × 205,481×321, 128×128 and 94×93, respectively. The originalimages of the test images are shown in Figure 3 and theircorresponding gray levels histograms are shown in Figure 4.

The threshold values obtained with the different thresh-olding methods on test images are shown in Table 1 and thecorresponding thresholded result by these threshold valuesfor test images are shown in Figures 5-8.

From Figures 5-8, it can be seen that the best segmentedresults can be obtained by the proposed method while thereare some defectives exist in the results by the other methods.For ‘tire’ image, the results obtained by Yen method andTao method are over-segmentation while keeping the under-segmentation error in the results obtained by Pun method,Benabdelkader method and Tang method. For ‘tire’ image,these five methods can not distinguish the object and back-ground well. For ‘plane’ image, the object is separated frombackground very well by the Tao method and the proposedmethod while a lot of background noise exist in the resultsobtained by other six methods, especially in the resultsobtained by Pun method, Benabdelkader method and Tangmethod. For ‘milkdrop’ image, there are same flaws exist inthe results by Pun method, Benabdelkader method and Tangmethod as that of in ‘plane’ image. For Otsu method, Kapurmethod and Yen method, there is much noise in the resultsby these three methods while for the Tao method, the over-segmentation error appears again in the segmented results.For ‘shot1’ image, the mentioned above flaws exist in theresults by Otsu method, Pun method, Benabdelkader method,Tao method and Tang method appear again while for theother three methods, the object separated from backgroundwell.

Because the effects of fuzzy correlation image segmen-tation depend on the selected threshold, it is essential tocompare the threshold selected by the DE method withthe threshold selected by the exhaustive search method.Comparison of results and time of the DE method andthe exhaustive search for test images is shown in Table2. Since DE is a stochastic global optimization algorithm,we repeated experiments to test the search stability of theDE method. Results and time of the DE method shownin Table 2 are the average results of 20 independent andrepeated experiments. As can be seen from Table 2, theDE method can give the similar optimal fuzzy parametercombination and threshold as that of the exhaustive search.In contrast, the search time of the DE method is very lessthan that of the exhaustive search. For example, the DEmethod and the exhaustive search obtain the similar optimalfuzzy parameter combination (0.0, 0.9, 255.0) and threshold75.0 for ‘tire’ image, but the search time of the DE methodis approximately one thousandth of that of the exhaustivesearch. Therefore, it can be concluded that the DE methodcan greatly improve the efficiency of the proposed imagesegmentation method. From the search results of all the ex-perimental images, we can conclude that the DE method hasgood adaptability in selecting the optimal fuzzy parametercombination and threshold when applied to different kindsof real-world images.

Time efficiency is an important index for different algo-rithms on image segmentation. So the experiments of CPUtime cost of all methods for obtaining the optimal thresholdvalue on every test image were also conducted in this paper.The results are reported in Table 3. The Otsu, Pun, Kapur etal., and Yen et al. methods are implemented by exhaustivesearch in this paper. The fast recursive algorithm was usedin Benabdelkader et al. method and Tang et al. method. Theant colony optimization algorithm was used in Tao et al.method. These three methods are implemented according totheir original paper in our experiment. From Table 3, it canbe seen that the exhaustive search algorithm used in Otsu,Pun, Kapur et al., and Yen et al. methods cost less CPUtime than other methods because the time complexity ofthese methods is O(L), where L is number of gray levels ofimage. The time complexity of other methods is larger thanO(L). Therefore, although some optimization algorithms orrecursive algorithms are used in these methods, it still costmuch CPU time than other methods. However, compared theproposed method with its exhaustive search version, it canbe seen from Table 2, the time cost of the proposed methodis less than that of the exhaustive search. For example, thesearch time of the proposed method is approximately onethousandth of that of the exhaustive search for test images.From Table 3, we can see that the cost time of the proposedmethod is also less than that of Tang et al. method.

Infrared target detection is widely application in manyfields [17], [21], [27], [28], such as video surveillance [29],[30]. At the last experiment, for further testing the perfor-mance of the proposed method, we applied the proposedmethod to infrared human image segmentation. Figure 9shows two infrared human images and their gray levels his-tograms. For convenience, we call these two infrared images‘IR1’ and ‘IR2’. Figures 10 and 11 show the segmentedresults by 8 different methods referenced in this paper.

From Figures 10 and 11, it can be seen that the humantargets are separated from background well by the proposedmethod while the results obtained by the other methodsare not ideal. For ‘IR1’ image, the thresholds obtained byOtsu method, Pun method, Benabdelkader method and Tangmethod are 67, 76, 76 and 68, respectively. When ‘IR1’image is segmented by these thresholds, the object can notbe distinguished from background. For Tao method, from ourexperiments, it usually will appear over-segmentation errorsometimes while for Kapur method and Yen method theirsegmentation is deficient at the some time. For example, theflaws of these methods appear in the segmented results of‘IR2’ image.

V. CONCLUSIONSIn this paper, a new image thresholding method is pre-

sented based on fuzzy partition and correlation theory. Atthe same time the DE algorithm is introduced into fuzzycorrelation image segmentation to find the best fuzzy pa-rameter combination of membership function and thresholdadaptively. Some real-world images and infrared humanimages are selected as the experimental data and the pro-posed method obtains satisfactory results in the segmentationexperiments. The DE method can obtain the similar optimalfuzzy parameter combination and threshold as that of theexhaustive search while its search time is very less than that



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(a) (b) (c) (d)

(e) (f) (g) (h)

Fig. 5: Segmented results of tire image by (a) Otsu method, (b) Pun method, (c) Kapur method, (d) Yen method, (e)Benabdelkader method, (f) Tao method, (g) Tang method and (h) The proposed method

(a) (b) (c) (d)

(e) (f) (g) (h)

Fig. 6: Segmented results of plane image by (a) Otsu method, (b) Pun method, (c) Kapur method, (d) Yen method, (e)Benabdelkader method, (f) Tao method, (g) Tang method and (h) The proposed method

(a) (b) (c) (d)

(e) (f) (g) (h)

Fig. 7: Segmented results of milkdrop image by (a) Otsu method, (b) Pun method, (c) Kapur method, (d) Yen method, (e)Benabdelkader method, (f) Tao method, (g) Tang method and (h) The proposed method



______________________________________________________________________________________

(a) (b) (c) (d)

(e) (f) (g) (h)

Fig. 8: Segmented results of shot1 image by (a) Otsu method, (b) Pun method, (c) Kapur method, (d) Yen method, (e)Benabdelkader method, (f) Tao method, (g) Tang method and (h) The proposed method

(a) (b)

(c) (d)

Fig. 9: Original infrared human images and their histogram. (a) IR1, (b) Histogram of IR1, (c) IR2 and (d) Histogram ofIR2.



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TABLE I: Threshold values obtained with the different thresholding methods on test imagesImage Otsu Pun Kapur Yen Benabdelkader Tao Tang Proposed

tire 84 20 110 83 21 180 25 75

plane 85 120 84 84 119 48 117 46

milkdrop 130 92 114 114 91 166 87 139

Shot1 174 147 131 131 164 68 149 128

TABLE II: Comparison of results and time of the DE method and the exhaustive search

ImageDE method Exhaustive search(a, b, c) Threshold Times(s) (a, b, c) Threshold Times(s)

tire (0.0, 0.9, 255.0) 75.0 0.2873 (0, 1, 255) 75.0 253.8

plane (0.0, 0.1, 157.8) 46.2 0.2361 (0, 1, 157) 46.3 186.7

milkdrop (1.0, 165.3, 233.9) 139.3 0.2437 (1, 165, 234) 139.2 215.2

Shot1 (117.5, 118.7, 154.2) 128.7 0.2795 (118, 119, 154) 128.9 193.7

TABLE III: Comparison of CPU time of the different methods (second)Image Otsu Pun Kapur Yen Benabdelkader Tao Tang Proposed

tire 0.0543 0.0517 0.0607 0.0496 0.0873 0.2791 2.3105 0.2873

plane 0.0419 0.0435 0.0431 0.0458 0.0791 0.2404 2.4307 0.2361

milkdrop 0.0426 0.0397 0.0409 0.0379 0.0765 0.2456 2.0172 0.2437

Shot1 0.0413 0.0462 0.0508 0.0437 0.0698 0.2798 1.9762 0.2795

(a) (b) (c) (d)

(e) (f) (g) (h)

Fig. 10: Segmented results of IR1 image by (a) Otsu method (t∗ = 67), (b) Pun method (t∗ = 76), (c) Kapur method(t∗ = 154), (d) Yen method (t∗ = 145), (e) Benabdelkader method (t∗ = 76), (f) Tao method (t∗ = 179), (g) Tang method(t∗ = 68) and (h) The proposed method (t∗ = 180)

of the exhaustive search. And the DE method has good searchstability in the repeated experiments. Therefore, the proposedmethod is an efficient image segmentation method and can beapplied to many fields, such as automatic target recognition,etc.

ACKNOWLEDGMENT

The authors would like to thank the anonymous refereesfor the valuable comments and suggestions which helped toimprove this paper.

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(a) (b) (c) (d)

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Fig. 11: Segmented results of IR2 image by (a) Otsu method (t∗ = 116), (b) Pun method (t∗ = 87), (c) Kapur method(t∗ = 103), (d) Yen method (t∗ = 103), (e) Benabdelkader method (t∗ = 89), (f) Tao method (t∗ = 154), (g) Tang method(t∗ = 87) and (h) The proposed method (t∗ = 106)

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Fangyan Nie was born in 1977 and graduated from National Universityof Defense Technology, Changsha, China, in 1999. He received his M.S.degree in Computer Science and Technology from Guizhou University,Guiyang, China, in 2005 and the Ph.D. degree in Instrument Scienceand Technology from Chongqing University, Chongqing, China, in 2010.His research interests include image processing, pattern recognition andintelligent optimization algorithms. He has published more than 20 papersin international/national journals.

Pingfeng Zhang was born in 1980 and graduated from Hunan Universityof Arts and Science, Changde, China, in 2008. Her research interestsinclude image processing, pattern recognition and intelligent optimizationalgorithms.



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Fuzzy Partition and Correlation for Image Segmentation ... · Fuzzy Partition and Correlation for Image Segmentation with Differential Evolution Fangyan Nie, Member, IAENG, and Pingfeng